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Modeling of pCO2 Point-of-Care Devices
byXuLiangLi
A thesis submitted to the Department of Chemical Engineering in conformity with the requirements for the degree of Master’s of Applied
Science
Queen’s University Kingston, Ontario, Canada
January, 2014 Copyright © Xu Liang Li, 2014
ii
AbstractA dynamic model is developed and presented that predicts the voltage response for a
Severinghaus electrode-based point-of-care pCO2 sensor. Eight partial differential
equations are derived to describe the diffusion and reaction phenomena in the sensor. The
model is able to predict the potential response versus time behaviour from different CO2
concentrations in the calibration fluid and control fluids.
The two most influential and uncertain parameters in the model are determined to be the
forward rate constant for benzoquinone consumption at the gold surface ( ), and the
partition coefficient for CO2 between the membrane and the electrolyte ( ). These
parameters were adjusted heuristically to obtain a good fit (within 2 mV) between the
dynamic voltage response data and the model predictions during a critical 4 second
period. The model predictions are sufficient for design sensitivity studies, however an
improved fit might be possible using a formal least-squares parameter estimation
approach, or if additional parameters were estimated.
Several design parameters are varied to study the influence of the electrolyte
concentration and the sensor geometry on the voltage response. The most influential
design parameter studied is the amount of water present in the electrolyte during sensor
operation. This can be affected by the amount of water evaporated during manufacturing
and storage, and by the amount of water present when the sensor “wets up” again during
operation. The amount of water picked up by the sensor in turn is affected by design
parameters such as component/membrane dimensions and thicknesses. The initial buffer
iii
concentration in the electrolyte is the second most influential parameter. The resulting
model can be used to perform “what if” analyses in order to understand the impact of
design decisions on the sensor performance, and to potentially improve the sensor from
performance and manufacturing cost perspectives.
iv
Acknowledgement First of all and foremost, I owe a big thank-you to my supervisors, Professor Kim McAuley and Professor Jim McLellan, for their guidance and delightful discussions during my Master’s degree. Kim and Jim, your kindness, patience, and expertise will be missed by me deeply. I will miss our time spent together in Dupuis Hall, Gordon Hall, and via Skype. I learnt so much about modeling and writing a thesis for the past two years. I would also like to thank my industrial supervisor, Dr. Craig Jeffrey, for his help with gathering all the data and discussions on how the sensor works. Thanks to Dr. Glenn Martin who was very patient to explain the electrochemistry at the electrodes. Special thanks to Stephen Snyder and Niko Lee-Yow for helping me with COMSOLTM implementation. I am indebted to the thesis examine committee, Dr. Brian Amsden, Dr. Dominik Barz and Dr. Jon Pharoah. They took the time to go through my thesis and gave me many valuable suggestions. Financial support from Queen’s University, MITACS, Ontario Centre of Excellence and our industrial sponsor Abbott Point of Care made this project possible and there was no way to do this project without them. My office mates, Jessica, Hui, Ryan, Abdullah, Hadis, Emily, Zahra and Yasmine made my time in Dupuis Hall G36 enjoyable and fun. Thanks for all the fun we had together especially with food and I will miss you all very much. Finally, I would like to thank my parents for encouraging me throughout my Master’s journey whenever I am excited or frustrated with my simulation results.
v
TableofContentsAbstract ............................................................................................................................... ii
Acknowledgement ............................................................................................................. iv
List of Tables ..................................................................................................................... vi
List of Figures ................................................................................................................... vii
List of Abbreviations .......................................................................................................... x
List of Nomenclature ......................................................................................................... xi
Chapter 1 Introduction ........................................................................................................ 1
Chapter 2 Model Development ......................................................................................... 15
2.1 Summary of Equation Development, Boundary Conditions, and Initial Model Parameters ..................................................................................................................... 19
2.2 Development of Material Balances and Boundary Conditions............................... 27
2.3 Model Implementation in COMSOLTM: ................................................................. 43
2.4 Summary ................................................................................................................. 58
Chapter 3 Simulation Results............................................................................................ 58
3.1 Overview ................................................................................................................. 58
3.2 Simulated Voltage Response Plots for Calibrant .................................................... 63
3.3 Identifying the Two Most Influential Parameters ................................................... 67
3.4 Sensitivity Analysis ................................................................................................ 73
3.5 Summary ................................................................................................................. 86
Chapter 4 Conclusions and Recommendations ................................................................. 88
4.1 Conclusions ............................................................................................................. 88
4.2 Contributions of this Thesis .................................................................................... 90
4.3 Recommendations for Future Work........................................................................ 91
References: ........................................................................................................................ 92
Appendix: .......................................................................................................................... 96
vi
ListofTablesTable 1: Model Assumptions ........................................................................................... 15 Table 2: Initial Parameters Used in Model Development. ................................................ 20 Table 3: Table of PDEs. .................................................................................................... 22 Table 4: Boundary and Initial Conditions ......................................................................... 23 Table 5: Heat of Reactions ................................................................................................ 31 Table 6: Values of α from 273.15 K to 348.15 K. ............................................................ 41 Table 7: Lumped Diffusion Coefficients Used in Model ................................................. 43 Table 8: Parameter List in COMSOLTM. .......................................................................... 45 Table 9: Variable List ....................................................................................................... 47 Table 10: pCO2 Values of Calibrant and Control Fluids. ................................................. 60 Table 11: Scaled Sensitivity Matrix for Parameter Ranking ............................................ 69
vii
ListofFiguresFigure 1: Traditional Severinghaus Carbon Dioxide Sensor. The electrolyte contains NaHCO3 and water. ............................................................................................................ 3 Figure 2: Potential change vs. time for a traditional Severinghaus sensor . ....................... 4 Figure 3: Improved pCO2 Sensor. The electrolyte contains NaHCO3, BQ, carbonic anhydrase, sucrose, KCl. ..................................................................................................... 7 Figure 4: Sensor operation timeline .................................................................................... 9 Figure 5: Change in potential difference vs. time during sensor operation. ..................... 10 Figure 6: Concentration profile in Ross Equation. ........................................................... 12 Figure 7: Side View of Sensor with Flow Channel. ......................................................... 19 Figure 8: Electrodes with electrical connection strips. ..................................................... 19 Figure 9: Differential element of height Δz and thickness Δr. ......................................... 27 Figure 10: Schematic of pCO2 Sensor .............................................................................. 36 Figure 11: Small ring with thickness Δr ........................................................................... 37 Figure 12: Curves of best fit for α and B. ......................................................................... 42 Figure 13: Tree structure of the Transport of Diluted Species Modules .......................... 45 Figure 14: Voltage response at the Au electrode .............................................................. 47 Figure 15: Sub tabs in Transport of Diluted Species modules .......................................... 50 Figure 16: Diffusion coefficients under “Diffusion GPM” and “Diffusion Electrolyte” . 51 Figure 17: Initial conditions in membrane and electrolyte ............................................... 53 Figure 18: Reaction tab in electrolyte ............................................................................... 54 Figure 19: Continuous flux at the interphase between gas-permeable membrane and electrolyte .......................................................................................................................... 55 Figure 20: BCs involving the Henry’s constant. ............................................................... 56 Figure 21: BCs involving the partition coefficient between the membrane and the electrolyte .......................................................................................................................... 57
viii
Figure 22: Reaction flux at the Au electrode. ................................................................... 58 Figure 23: Voltage responses as a function of time for calibrant (from t2 to t3) and control fluids (from t4 to t5) for a) CV1, b) L2, c) GB, and d) CV5. Green error bars for measurements are show at two points to illustrate reproducibility. .................................. 63 Figure 24: Voltage response for a simulation for calibrant in COMSOLTM with kfAu = 666.28 ∙ ......................................................................................................... 64 Figure 25: Voltage response for a simulation with kfAu = 160 ∙ .................... 65 Figure 26:Voltage response for a simulation with kfAu = 700 ∙ ..................... 66 Figure 27: Voltage response for all control fluids. The red marks are data points from experiments. The solid red marks are the initial predetermined number of seconds that are used to calculate the mean voltage response. .................................................................... 66 Figure 28: Mean voltage response vs pCO2 plots for the base and tuned cases ............... 71 Figure 29: Voltage response for all control fluids for using initial parameter values in Table 8 are represented by dashed lines and voltage response using tuned parameter values are represented by solid lines. ................................................................................ 72 Figure 30: Concentration profiles for carbonic acid at the middle of the Au surface. ..... 72 Figure 31: Voltage response for calibrant used for concentration profiles at centre of the Au electrode ...................................................................................................................... 73 Figure 32: Centre of Au electrode where carbonic acid concentration profiles are taken 73 Figure 33: Influence of water concentration in electrolyte voltage response for the calibrant............................................................................................................................. 75 Figure 34: Influence of water concentration in electrolyte on voltage responses for control fluids. ................................................................................................................................ 76 Figure 35: Influence of the electrolyte height on voltage response for the calibrant. ...... 77 Figure 36: Influence of water concentration in electrolyte on voltage responses for control fluids ................................................................................................................................. 77 Figure 37: Influence of water concentration in electrolyte and electrolyte thickness on voltage response for the calibrant. .................................................................................... 78
ix
Figure 38: Influence of water concentration in electrolyte on voltage responses for control fluids ................................................................................................................................. 79 Figure 39: Influence of initial carbonic acid concentration in electrolyte on voltage response for the calibrant. ................................................................................................. 80 Figure 40: Influence of initial carbonic acid concentration in electrolyte on voltage responses for control fluids ............................................................................................... 81 Figure 41: Influence of initial buffer concentration in electrolyte on voltage response for the calibrant. ...................................................................................................................... 82 Figure 42: Influence of initial buffer concentration in electrolyte on voltage responses for control fluids ..................................................................................................................... 83 Figure 43: Influence of initial BQ concentration in electrolyte on voltage response for the calibrant............................................................................................................................. 84 Figure 44: Influence of BQ concentration in electrolyte on voltage response for control fluids ................................................................................................................................. 84 Figure 45: Influence of Au electrode surface area voltage response for the calibrant ..... 85 Figure 46: Influence of Au electrode surface on voltage responses for control fluids. .... 85
x
ListofAbbreviationsPOC Point-of-care
PDE Partial differential equation
BC Boundary conditions
xi
ListofNomenclature Symbol Description kf forward rate constant kr reverse rate constant E electrode potential of gold Eo standard electrode potential of gold R ideal gas constantT absolute temperaturen stoichiometric number of electrons transferredF Faraday constantt Time L thickness of electrolyte layer in Ross model Ce(0) the initial centration of H2CO3 in the electrolyte
Ce() final concentration that would be reached when the concentration of H2CO3 in the electrolyte becomes equal to the concentration of H2CO3 in the blood sample
D diffusion coefficient driving force in the Ross model
Z vertical axis R radial direction R1 Width of electrolyte R2 Width of membrane Z1 Height of electrolyte Z2 Height of membrane RAuI Inner radius of Au electrode RAuO Outer radius of Au electrode a mean radius of Cl- Diffusion coefficient of CO2 in membrane Diffusion coefficient of Diffusion coefficient of Diffusion coefficient of Diffusion coefficient of Diffusion coefficient of
Diffusion coefficient of Diffusion coefficient of
Equilibrium constant of Equilibrium constant of Equilibrium constant of O Equilibrium constant of the electrochemical reaction at Au electrode surface
Initial concentration of Initial concentration of Initial concentration of
xii
Initial concentration of Initial concentration of Initial concentration of
Initial concentration of Initial concentration of Initial concentration of
Initial concentration of Initial concentration of
H Henry’s constant Partition coefficient
Forward reaction rate constant at the Au electrode
Forward reaction rate constant for the dissociation of carbonic acid
Forward reaction rate constant for the dissociation of bi carbonate
Forward reaction rate constant for the dissociation of water
Reverse reaction rate constant at the Au electrode Reverse reaction rate constant for the dissociation of bicarbonate
Reverse reaction rate constant for the dissociation of carbonic acid
Reverse reaction rate constant for the dissociation of water
Chapter1Introduction
The global market for Blood Gas and Electrolyte Analyzers has been forecasted at 27,432
units and US$477 million by the year 2017 (Global Industry Analysts, Inc., 2012). The
reasons for this substantial demand are that the aging population will lead to more
patients demanding critical care, and the growing popularity of point-of-care (POC)
blood gas testing due to its simplicity, rapid read-out, and portability. One important test
that is routinely performed using POC technology is pCO2, which determines the partial
pressure of dissolved carbon dioxide in the blood. pCO2 is an indicator of the acid/base
chemistry of the human body and is therefore one important blood test for assessing the
condition of patients (Lane and Walker, 1987).
The Abbott Point of Care pCO2 POC sensor is a potentiometric sensor based on a
Severinghaus design (Severinghaus and Bradley, 1958; Lauks and Maczuszenko, 2006;
Davis et al., 1996). In this sensor, CO2 diffuses from the blood sample through a
membrane and into the aqueous electrolyte. The sensor detects the resulting pH change
associated with the dissolution of CO2 and the formation of carbonic acid. Changes in
pCO2 result in changes in [H+] at the cathode surface. In traditional Severinghaus sensors
there is a pH-sensitive glass electrode that detects the pH change in the electrolyte. The
change in pH is governed by the chemical equilibrium (Severinghaus and Bradley, 1958):
⇌ (1)
2
The traditional Severinghaus sensor (generally not employed in POC applications) is
composed of a pH electrode, a reference electrode (usually a Ag/AgCl electrode), and a
NaHCO3 electrolyte buffer solution, which is separated from the sample by a gas-
permeable membrane. NaHCO3 in the electrolyte is completely dissociated into Na+ and
HCO-3. These additional HCO-
3 ions affect the pH by shifting equilibrium reaction (1) to
the left. High concentrations of bicarbonate (e.g., 2 mmol/L) influence the sensitivity of
the sensor to CO2 and lengthen the sensor’s response time (Zosel et al., 2011) which is
undesirable. Figure 1 shows a traditional Severinghaus carbon dioxide sensor (Ross et
al., 1973).
3
Figure 1: Traditional Severinghaus Carbon Dioxide Sensor (after Ross et al., 1973;
Davis et al., 1996). The electrolyte contains NaHCO3 and water.
The blood sample at the bottom is in contact with the gas-permeable membrane, which
permits some of the CO2 to diffuse into the electrolyte, until equilibrium is reached.
The potential of the pH glass electrode changes relative to the reference electrode as the
acidity changes due to the dissolved CO2.
One shortcoming of the traditional Severinghaus sensor is that equilibrium needs to be
established between the CO2 level in the sample and the electrolyte so that accurate
readings can be obtained. Many articles concerned with the design of Severinghaus
electrodes focus on shortening the response time for this equilibrium to be established
4
(Zhao and Cai, 1997; Tongol et al., 2003; Meyerhoff et al., 1983; Lopez, 1984). Also, the
traditional Severinghaus sensor has difficulties when measuring low CO2 levels, due to
poor sensitivity and to CO2 depletion from the sample as CO2 diffuses into the
electrolyte. Therefore, achieving wide detection limits has been an important issue in
Severinghaus sensor design (Cai and Reimers, 1993). Severinghaus sensors have been
modified by many scientists to achieve improved response time and accuracy (Zhao and
Cai, 1997; Tongol et al., 2003; Meyerhoff et al., 1983; Lopez, 1984).
Figure 2 shows the dynamic response of a traditional Severinghaus sensor when it is used
for measuring CO2 concentrations in ocean water (Cai and Reimers, 1993). Steady-state
data obtained after 40 minutes would be used to calculate pCO2.
Figure 2: Potential change vs. time for a traditional Severinghaus sensor (from Cai and Reimers, 1993).
5
In the improved Severinghaus sensor for POC usage that is modeled in this study, the pH
glass electrode is replaced by a gold electrode (Davis et al., 1996; Lauks and
Maczuszenko, 2006). As in the traditional design, when the sample is in contact with the
sensor, CO2 diffuses through a gas-permeable membrane and into the electrolyte, which
is in contact with a Ag/AgCl reference electrode and the gold electrode. The dissolved
CO2 reacts with water to form carbonic acid, which dissociates to form hydrogen ions as
shown in reaction (1). The improved sensor contains sodium bicarbonate and carbonic
anhydrase enzyme, which catalyzes the hydration of CO2 to speed up the sensor response
time and reduce sensor drift (Zhao and Cai, 1997; Zosel et al., 2011). The electrolyte
also contains benzoquinone (BQ) ( Lauks, 1998; Lauks and Maczuszenko, 2006; Davis et
al., 1996). The potential of the Au electrode changes when BQ reacts with hydrogen ions
and electrons at the Au electrode surface (Hui et al., 2009):
BQ + 2e- + 2H+ ⇌ H2Q (2)
Changes in the potential difference over time are measured between the Au electrode and
the reference Ag/AgCl electrode. Reaction (2) is actually a series of elementary reactions
(Guin et al., 2011):
BQ + e- ⇌ BQ˙- (3)
BQ˙- + H+ ⇌ H˙Q (4)
H˙Q + e- ⇌ HQ- (5)
HQ-+H+ ⇌ H2Q (6)
6
If reaction equilibrium is assumed to exist at the Au/electrolyte surface, the Nernst
equation can be used to determine the change in potential based on concentrations of the
reactants (Hui et al., 2009):
(7)
where:
: electrode potential of the quinone reaction (V)
: standard electrode potential of the quinone reaction which is 0.699 V (Dabos,
1975)
R: ideal gas constant (8.314
)
T: absolute temperature (K)
n: stoichiometric number of electrons transferred, which is 2 in this case
F: Faraday constant (96,485.3365 )
If the Nernst equation is used (i.e., equilibrium is assumed for reaction (2)), then
knowledge of the individual rates for reactions (3) to (6) is not required.
The improved sensor that is modeled in this study is a very small lab-on-a-chip device
with the width of the Au electrode being approximately 10 microns (Davis et al., 1996).
7
Figure 3 is a schematic diagram of the POC pCO2 sensor.
Figure 3: Improved pCO2 Sensor. The electrolyte contains NaHCO3, BQ, carbonic anhydrase, sucrose, KCl (after Davis et al., 1996; Lauks and Maczuszenko, 2006).
In addition to the sodium bicarbonate, BQ and carbonic anhydrase, the electrolyte also
includes sucrose and KCl. The function of the sucrose is to keep the water within the
electrolyte solution (i.e., via osmotic pressure), rather than having it diffuse out through
the water- and CO2- permeable membrane over time. The KCl affects the activity of the
Cl- in the electrolyte which in turn changes the potential of the reference Ag/AgCl
electrode due to the ionic strength of the electrolyte. The activity of Cl- is discussed in
depth in Chapter Two.
The measured voltage from experimental data is governed by Equation (8):
(8)
where is the measured voltage (V), is the electrode potential of the quinone
reaction (V) as shown in Equation (7), and is the voltage at the reference Ag/AgCl
electrode (V).
8
The KCl in the electrolyte influences the potential of Ag/AgCl electrode. The reference
Ag/AgCl electrode potential is calculated via Equation (9):
/ ln (9)
where / is the electrode potential of the reference Ag/AgCl electrode (V), which
is 0.22233 V at 25 oC when compared against the standard hydrogen electrode (SHE)
(Greeley and Smith et. al., 1960). is the activity or effective concentration of
chloride ion, Cl-, in , which is discussed in Chapter Two.
Substituting Equations (7) and (9) into Equation (8) gives:
0.699 ln 0.22233 ln
(10)
This equation will be used in the model developed in this thesis to relate concentrations
of H2Q, BQ and H+ to the measured voltage.
The i-STAT POC test system produced by Abbott Point of Care consists of a portable
handheld analyzer that is capable of running various test cartridges for species of clinical
interest such as creatinine, Na+, pCO2, and etc. Each cartridge containing the pCO2 sensor
is constructed with an onboard calibrant pack containing the calibration fluid. The
operator fills the sample inlet well on each cartridge with a sample to be tested, seals the
cartridge and inserts the cartridge into the handheld i-STAT analyzer to commence
testing of the sample. When the cartridge is inserted into the analyzer, the calibrant and
sample fluids are heated to 37 ºC (human body temperature) during the course of the test.
The sensor in Figure 3 is calibrated with each run (i-STAT cartridges are single-use
9
devices) so that reliable and accurate readings can be obtained. Note that the time scale
has been normalized. A calibration fluid (aqueous solution with a known CO2
concentration) is first delivered to the upper surface of membrane (before the blood
sample is delivered). After a brief period of heating, the calibrant fluid comes in contact
with the membrane so that CO2 from the calibration fluid can begin diffusing into the
electrolyte. This initial contact time is denoted as 0 seconds in Figure 4.
Figure 4: Sensor operation timeline (after Cozzette et a.l, 1992; Davis, 1998)
Heating to 37 oC is complete at 0.88, after which instrument and assay diagnostic checks
are conducted for a brief period before data collection begins. At 1.15, readings of the
potential difference between the Ag/AgCl electrode and the Au electrode are recorded
until 1.48, when the calibration fluid is replaced by the blood sample, which induces a
different dynamic response in the potential. Reliable readings of the potential are
available at 1.57 from start of the “sample push”. The corresponding voltage response in
Figure 5 and the known CO2 concentration in the calibrant are then used to calculate the
CO2 partial pressure in the blood sample using an empirical model. Note that several
values from the potential vs. time curves, rather than the final steady-state value (as used
by the traditional Severinghaus sensor) are used to compute the CO2 concentration in the
blood sample.
10
Figure 5: Change in potential difference vs. time during sensor operation (after Cozzette et a.l, 1992; Davis, 1998)
Until now, the only fundamental mathematical models that have been developed to
describe potentiometric sensors for CO2 are based on the work of Ross et al. (1973) who
developed a simplified model to describe the operation of a traditional Severinghaus
pCO2 sensor (Jensen and Rechnitz, 1979; Cai and Reimers, 1993; Zosel et al., 2011).
The Ross model was developed to determine the influence of various design parameters
that affect the time t required to achieve a certain fractional approach to equilibrium for
the aqueous CO2 (i.e., the H2CO3) in the electrolyte. The Ross model:
(11)
11
is useful for determining when the sensor response will be sufficiently close to steady
state so that reliable measurements can be made using a traditional Severinghaus sensor
(see Figure 2). The situation described by the Ross model is shown in Figure 6 where
Ce(0) is the initial centration of H2CO3 in the electrolyte and Ce() is the final
concentration that would be reached when the concentration of H2CO3 in the electrolyte
becomes equal to the concentration of H2CO3 in the blood sample. L is the thickness of
the electrolyte layer, m is the thickness of the gas-permeable membrane, D is the
diffusion coefficient of CO2 in the gas-permeable membrane and k is a partition
coefficient. The Ross model assumes that there is a linear concentration gradient across
the gas-permeable membrane and Equation (12) is developed by assuming that the
diffusion rate across the membrane is equal to the rate of accumulation of aqueous CO2
(and related ionic species) within the electrolyte.
12
Figure 6: Concentration profile in Ross Equation (after Zoel et al., 2011).
In the Ross model ϵ is the dimensionless driving force:
∞∞
(12)
that approaches zero as H2CO3 accumulates in the electrolyte. (the change in the
concentration of carbonic acid’s associated ionic species over the change in the
concentration of H2CO3) accounts for the accumulation of ionic species (HCO3- plus
CO32-) that are in equilibrium with H2CO3. This ratio is required in the model because
only a fraction of the CO2 that diffuses through the membrane accumulates as H2CO3.
The value of can be assumed to be constant when the concentration of H2CO3 is high
(i.e., higher than 2 mmol/L) (Zoel et al., 2011).
13
Assumptions in the Ross model include:
1. Cb()=Cb(0) is constant because the blood volume is large compared to the
electrolyte in Figure 6.
2. At the final steady state Cb() = Ce() in Figure 6.
3. There is a linear CO2 gradient within the gas-permeable membrane.
4. Ce(t) is spatially uniform in the electrolyte because the only important
resistance to mass transfer is in the gas-permeable membrane depicted in
Figure 6.
5. All carbonate species in the electrolyte are in equilibrium.
6. The pH electrode responds instantaneously to pH changes in the electrolyte
solution, which are caused by changes in Ce(t) and the associated changes in
the concentrations of ionic species.
7. The partition coefficient for CO2 between the blood and the membrane is the
same as the partition coefficient for CO2 between the electrolyte and the
membrane.
In the proposed model for the POC pCO2 sensor, assumptions 1, 3 and 4 will be relaxed.
Assumption 6 does not apply because there is no pH electrode in the modified
Severinghaus system that will be modeled.
The Ross model only accounts for the diffusion of CO2 through the gas-permeable
membrane, neglecting concentration gradients of CO2 and other species within the
electrolyte. The model developed in this thesis will not only account for the CO2 in the
14
gas-permeable membrane but will also account for the chemical reactions that occur in
the electrolyte and at the electrode surface.
To my knowledge, the only model of a Severinghaus system that exhibits greater
complexity than the Ross model is that developed by Samukawa et al. (1995). The
system modeled consists of a pH electrode at the centre of a cylindrical container of
electrolyte, which is surrounded by a cylindrical gas permeable membrane. Samukawa et
al. used partial differential equations (PDEs) in cylindrical coordinates to account for
changes in the partial pressure of dissolved CO2 in time and two spatial dimensions.
Zero-diffusion boundary conditions (BCs) were specified at the upper and lower surfaces
of the electrolyte and at the surface between the electrolyte and the electrode.
Presumably, the other BC in the radial direction is that the partial pressure of CO2 at the
outer edge of the membrane is equal to the partial pressure of CO2 within the sample.
Samukawa et al. used diffusion coefficients of 10-12 m
2/s and 10-9 m
2/s, respectively, for
CO2 in the gas-permeable membrane and the electrolyte. The PDEs were discretized and
solved using finite difference approximations and simulation results were shown for a
variety of CO2 pressures in the sample.
Like the model of Samukawa et al., the model proposed in this thesis will also use PDEs
to model diffusion of CO2 and H2CO3 in a Severinghaus sensor. The proposed model
will be more detailed and will account for additional species (i.e., HCO , H , OH , Na ,
,CO , BQ and H2Q) and for the different geometry used in the POC sensor.
15
The purpose of the current research project is to develop a mathematical model that will
allow a POC pCO2 sensor manufacturer to better estimate the effects of sensor design
parameters on sensor performance. The proposed model will describe the potential
versus time behaviour that results from the different CO2 concentrations in the calibration
fluid and standardized samples that are used for quality assurance. Many parameters in
the model can be varied to influence the sensor performance including the dimensions of
the membrane, electrolyte and electrodes and the concentrations of various species.
In this thesis, Chapter Two is concerned with model development and model
implementation in COMSOLTM, including details such as setting up the BCs at the Au
electrode surface. Chapter Three provides the simulation results and describes the
methods used for parameter tuning. Conclusions and recommendations are provided in
Chapter Four.
Chapter2ModelDevelopment The mathematical model for the POC pCO2 sensor consists of a set of material balances
on the physical elements of the sensor. A simplified geometry is assumed for the
membrane and electrode shown in Figures 7 and 8, so that the model equations can be
written using cylindrical coordinates. Note that any effects of the electrode strips (see
Figure 8 and Assumption 1.1 in Table 1) are ignored so that the behaviour of the sensor is
uniform in the angular θ direction. The full set of assumptions used to develop the
mathematical model is provided in Table 1.
Table 1: Model Assumptions
16
1.1 Concentrations of diffusing species in the membrane and electrolyte
change in the vertical (z) direction and the radial (r) direction. There
are no concentration gradients in the angular direction (θ), and the
impact of gap in annular shape of the Au electrode and the electrode
strips (see Figure 8) is negligible.
1.2 The CO2 partial pressure in the blood is uniform and at the bulk value.
The concentration of CO2 within the blood remains constant over time
because the amount of CO2 depletion is negligible.
1.3 The gas-permeable membrane is permeable to CO2 but not to other
species in the blood or electrolyte. Any transfer of water through the
membrane is neglected.
1.4 There is no bulk flow in this enclosed system.
1.5 The electrolyte has been heated to 37 oC at time zero so that changes
in diffusivities and equilibrium constants over time due to temperature
changes can neglected. Other factors that may influence diffusivities
or equilibrium constants are also neglected, except for the influence of
voltage on the equilibrium constant KAu Surface for the reaction at the
surface of the Au electrode, which is included via Equation (31).
1.6 Concentrations within the electrolyte are spatially uniform at time
zero. Any changes in water concentration within the electrolyte with
time or position are negligible due to the high water concentration.
1.7 When the electrolyte solution is manufactured, it contains
17
significantly more water than the “dried down” electrolyte contained
in the sensor. The amount of water that evaporates during sensor
production and storage is not well known. In base case simulations,
half of the water in the “as manufactured” electrolyte solution is
assumed to be present in the electrolyte so that the concentrations of
BQ and the NaHCO3 buffer are twice of those in the “as
manufactured” electrolyte. Other simulations are conducted using
different amounts of water.
1.8 The initial H2CO3 concentration in the electrolyte during
manufacturing is assumed to be 10-5 mol L-1 in the base case
simulations. This concentration would be in equilibrium with air at 25
ºC containing a typical CO2 concentration of 0.04 kPa (Manahan,
2005). The initial H2CO3 concentration is changed in other
simulations.
1.9 Sucrose does not participate in any chemical reactions in the
electrolyte. Its only influence is to help keep the water concentration
within the electrolyte constant during sensor storage and use.
1.10 NaHCO3 is assumed to dissociate completely. The Cl- ions from the
entirely dissociated KCl in the electrolyte influence the potential of
the reference Ag/AgCl electrode, but the K+ ions have no important
influence on the operation of the sensor.
1.11 The potential of the Ag/AgCl electrode is constant during the sensor
operation.
18
1.12 The sensor is able to replace the calibration fluid with blood sample
instantaneously at the membrane surface. There is no influence of the
air bubble between the calibration fluid and the blood sample as it
passes quickly over the membrane surface.
1.13 Diffusion occurs due to concentration gradients alone. Potential
gradients within the electrolyte are small because they are not
imposed, but arise from ion diffusion. As a result, migration of ions
due to potential gradients is neglected. Cations and anions in the
electrolyte diffuse in pairs or threesomes to maintain electroneutrality,
which influences the values of the lumped diffusivity parameters
chosen for use in the model (see Table 7).
1.14 The Ag/AgCl and Au electrodes protrude into the electrolyte by only a
small distance (see Figure 7) so that the bottom boundary of the
electrolyte can be treated as a flat surface in the model.
1.15 The degrees of polymerization of the benzoquinone and hydroquinone
in the electrolyte have no influence on the operation of the sensor
(Lindsey, 1974).
1.16 H+, BQ and H2Q are the only species that influence the potential at the
surface of the Au electrode.
1.17 There is a small surface current at the Au electrode, but there is
negligible external current. The forward rate constant kfAu is assumed
to be constant and is not influenced by the small changes in voltage
that are detected. However, the reverse rate constant krAu, which is
19
calculated from Equations (29) does depend on the voltage because
the equilibrium constant KAu Surface is voltage dependent.
Figure 7: Side View of Sensor with Flow Channel (after Davis, 1998; Lauks, 1998).
Figure 8: Electrodes with electrical connection strips (after Davis, 1998; Lauks, 1998).
2.1SummaryofEquationDevelopment,BoundaryConditions,andInitialModelParameters
Parameters used in the model, along with their initial literature values, are in shown in
Table 2. Note that any parameter regarding the concentration of a particular species has a
unit of mol/L due to the fact that Abbott Point of Care uses this unit for their stock
solutions. However, this unit is converted to mol/m3 for the derivation of PDEs and
20
model implementation in COMSOLTM. Material balance partial differential equations
(PDEs) developed for CO2 in the membrane and H2CO3, HCO , H , OH ,CO ,BQ, and
H2Q in the electrolyte are provided in Table 3. Initial and boundary conditions for the
PDEs are provided in Table 4, where the conditions are grouped according to the
corresponding PDEs (e.g., conditions (4.1.1) to (4.1.7) correspond to PDE (3.1)).
Representative derivations for selected PDEs and BCs are provided in Section 2.2 along
with a discussion of some of the parameter values. The derivations presented illustrate the
approach taken (and the issues encountered) in developing the PDE model and BCs.
Table 2: Initial Parameters Used in Model Development.
Symbol Description Initial Value and Range if Applicable
Reference
R1 Width of electrolyte 2.80 10 m APOC Sensor R2 Width of membrane 2.816 10 m APOC Sensor Z1 Height of electrolyte 2.05 10 m APOC Sensor Z2 Height of membrane 3.65 10 m APOC Sensor RAuI Inner radius of Au electrode 10.2 10 m APOC Sensor RAuO Outer radius of Au electrode 1.55 10 m APOC Sensor a Mean radius of Cl- 0.78 nm (Raghunathan and Aluru,
2006) Diffusion coefficient of CO2 in
membrane at 37 oC 2.20 0.2 10 m2/s
(Yang and Kao, 2010)
Diffusion coefficient of at 37 oC
2.70 0.2 10 m2/s
(Zeebe, 2011)
Diffusion coefficient of at 37 oC
1.57 0.2 10 m2/s
(Newman and Thomas-Alyea, 2004)
Diffusion coefficient of at 37 oC
1.25 0.2 10 m2/s
(Newman and Thomas-Alyea, 2004)
Diffusion coefficient of at 37 oC
7.04 0.2 10 m2/s
(Newman and Thomas-Alyea, 2004)
Diffusion coefficient of at 37 oC
1.08 0.2 10 m2/s
(Zeebe, 2011)
21
Diffusion coefficient of at 37 oC
1.18 0.2 10 m2/s
(Green and Perry, 2008)
Diffusion coefficient of at 37 oC
1.18 0.2 10 m2/s
Green and Perry, 2008)
Equilibrium constant of at 37 oC
6.30 0.5
10
(Snokeyink and Jenkins, 1980) (Haynes, 2012)
Equilibrium constant of at 37 oC
5.78 0.5
10
(Snokeyink and Jenkins, 1980) (Haynes, 2012)
Equilibrium constant of O at 37 oC
2.39 0.5
10
(Snokeyink and Jenkins, 1980) (Haynes, 2012)
Initial concentration of APOC Company Confidential
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H Henry’s constant at 37 oC 0.230 0.002
∙
(Burtis et al., 2006)
Partition coefficient at 37 oC 1 0.5 (Burtis et al., 2006)
Forward reaction rate constant at the Au electrode at 37 oC
505 495
∙
Estimated
22
Forward reaction rate constant for the dissociation of carbonic acid at 37 oC
1000 Assumed to be large
Forward reaction rate constant for the dissociation of bi carbonate at 37 oC
1000 Assumed to be large
Forward reaction rate constant for the dissociation of water at 37 oC
1000 Assumed to be large
Reverse reaction rate constant for the dissociation of bicarbonate at 37 oC
Due to equilibrium ratio
Reverse reaction rate constant for the dissociation of carbonic acid at 37 oC
Due to equilibrium ratio
Reverse reaction rate constant for the dissociation of water at 37 oC
Due to equilibrium ratio
3 Concentration of bicarbonate before time 0 0.050
molL
(Abbott Point of Care-d, 2012)
Concentration of BQ before time 0
(Abbott Point of Care-d, 2012)
Concentration of before time 0 1 10
molL
(Manahan, 2005)
Density of water at 37 oC 0.993333 (Colt, 2012)
Table 3: Table of PDEs. 3.1 1
3.2 1
23
3.3
1
3.4 =
3.5 =
3.6 1
3.7
1
3.8
1
Table 4: Boundary and Initial Conditions
4.1 4.1.1 | ,
4.1.2 ,
| ,
4.1.3 ,
0
4.1.4 | , (note that H is the Henry’s law constant)
4.1.5 ,
|
24
4.1.6 ,
0
4.1.7 , ,
4.2 4.2.1 , ,
4.2.2 ,
0
4.2.3 , ,
4.2.4 ,
0
4.2.5 , , 4.3 4.31
,
0
4.3.2
,
0
4.3.3 ,
0
4.3.4 ,
0
4.3.5 0, ,
4.4 4.4.1 ,
0
4.4.2 ,
0
4.4.3 ,
2
4.4.4 ,
0
25
4.4.5 ,
0
4.4.6 ,
0
4.4.7 , ,4.5 4.5.1
,0
4.5.2 ,
0
4.5.3 ,
0
4.5.4 ,
0
4.5.5 , , 4.6 4.6.1
,0
4.6.2
,
4.6.3 ,
0
4.6.4
, 0
4.6.5 , , 4.7 4.7.1
,0
4.7.2
,
0
4.7.3 ,
4.7.4 ,
0
26
4.7.5 ,
0
4.7.6 ,
0
4.7.7 0, ,
4.8 4.8.1 ,
0
4.8.2 ,
0
4.8.3 ,
4.8.4 ,
0
4.8.5 ,
0
4.8.6 ,
0
4.8.7 0, ,
27
2.2DevelopmentofMaterialBalancesandBoundaryConditions The material balance equations – whether for species in the membrane, or the electrolyte,
can be derived by considering an annular element of height Δz and thickness Δr as shown
in Figure 9. The following material balance (in moles) can be written for any species of
interest in this control volume (either within the membrane or within the electrolyte):
Accumulation= in – out + generated – consumed
+ migration due to potential
(13)
Figure 9: Differential element of height Δz and thickness Δr.
Due to Assumption 1.13, the term migration due to potential is neglected. The only
species whose concentration needs to be modeled within the gas-permeable membrane is
the dissolved CO2. Even though water can also permeate the membrane, it is assumed
that there is no appreciable transport of water across the membrane and that the water
concentration in the electrolyte is constant (i.e., see Assumptions 1.6 and 1.7 in Table 1).
In the membrane, CO2 diffuses in at the outer surface of the annulus (i.e., at r+Δr) and out
at the inner surface (i.e., at r). CO2 also diffuses in at the upper surface (i.e., at z+Δz) and
out at the lower surface (i.e., at z). No reactions occur within the small element, so there
is neither generation nor consumption of CO2. Let [CO2]m be the concentration of
28
dissolved CO2 in mol/m3 in the membrane and be the corresponding diffusivity in
m2/s. The number of moles of CO2 that diffuse into the control volume at r+Δr during a
short period of time Δt is:
2 ∆ ∆∆
∆
The number of moles of CO2 that diffuse out at r is:
2 ∆ ∆
The number of moles diffusing in at z+Δz is:
2 ∆∆
∆
The number of moles diffusing out at z is:
2 ∆ ∆
The number of moles of CO2 that accumulate within the control volume within the period
Δt is:
2 | ∆ |
As a result, Equation (13) becomes:
2 | ∆ | (14)
29
2 ∆ ∆∆∆ 2 ∆ ∆
2 ∆∆∆ 2 ∆ ∆
Dividing both sides by 2 gives:
∆
∆
∆∆
∆ ∆
∆
∆
(15)
Taking the limit as Δt, Δr, Δz approach 0 gives:
(16)
which can be manipulated to give PDE (3.1) in Table 3.
The concentrations of all chemical species H CO , HCO , H , OH ,CO ,BQ, and
H2Q) in the electrolyte are also tracked using PDEs and algebraic equations to model the
transport of these species. Diffusion is assumed to be only due to concentration
gradients, and not due to potential gradients (i.e., see Assumption 1.13 of Table 1). These
material balance PDEs differ from the membrane balance by the presence of reactions
taking place in the electrolyte.
30
The reactions in the electrolyte are governed by the following chemical equilibria and
their equilibrium constants at standard conditions are well known (Maas et.al., 1971;
Snokeyink and Jenkins, 1980; Wimberley et.al., 1985):
⇌
(17)
⇌
(18)
O⇌
(19)
O
Equilibrium constants at 37 oC can be determined from the integrated form of the van’t
Hoff equation using the values at the standard conditions:
∆ 1 1
(20)
31
where R is the universal gas constant 8.314 ∙
, Tref is at 298.15 K, and T is 310.15
K.∆ is the standard heat of reaction in kJ/mol at 25 oC and 1 atm and the values of
∆ s for equilibrium reactions (17) to (19) are listed in Table 5(Haynes, 2012). The
reactions are endothermic.
Table 5: Heat of Reactions
Equilibrium Reactions ∆ (kJ/mol)
⇌
(17) 14.7
⇌ (18)
9.15
O
⇌ (19) 55.8
The resulting values at 37 ◦C are provided in Table 2 (Standard, 2012; Manahan, 2005).
The equilibrium constants in Equations (17) to (19) are ratios of the forward and reverse
rate constants for these dissociation reactions:
(21)
(22)
(23)
In the electrolyte, H2CO3 (which is CO2 dissolved in the aqueous phase) enters the outer
surface of an annulus at r+Δr and leaves the inner surface at r. H2CO3 also diffuses in at
32
the upper surface (i.e., at z+Δz) and out at the lower surface (i.e., at z). H2CO3 is also
consumed and generated via equilibrium reaction (1). Let [H2CO3] be the concentration
of dissolved CO2 in mol/m3 in the electrolyte and be the corresponding diffusivity
in m2/s. The number of moles of H2CO3 that diffuse into the control volume at r+Δr
during a short period of time Δt is:
2 ∆ ∆∆
∆
The number of moles of H2CO3 that diffuse out at r is:
2 ∆ ∆
The number of moles diffusing in at z+Δz is:
2 ∆∆
∆
The number of moles diffusing out at z is:
2 ∆ ∆
The number of moles of consumed is:
2 ∆
The number of moles of generated is:
33
2 ∆
The number of moles of H2CO3 that accumulate within the control volume during the
period Δt is:
2 | ∆ |
As a result, the material balance Equation (13) for in the electrolyte becomes:
2 | ∆ |
2 ∆ ∆∆
∆
2 ∆ ∆
2 ∆∆
∆
2 ∆ ∆
2 ∆
2 ∆
(24)
Dividing both sides by 2 gives:
The number of moles of H2CO3 that accumulate within the control volume during the
period Δt is:
2 | ∆ |
34
As a result, the material balance Equation (13) for in the electrolyte becomes:
2 | ∆ |
2 ∆ ∆∆
∆
2 ∆ ∆
2 ∆∆
∆
2 ∆ ∆
2 ∆
2 ∆
(25)
Dividing both sides by 2 gives:
| ∆ |
∆
∆
∆
∆ ∆
∆
∆
(26)
Taking the limit as Δt, Δr, Δz approach 0:
35
(27)
which is being manipulated to Equation (3.2) in Table 3.
The material balance PDEs for the remaining six species in the electrolyte were derived
in an analogous fashion and are listed as Equations (3.3) to (3.8) in Table 3. Note that
since there are no reactions of BQ and H2Q in the bulk of the electrolyte, their PDEs
resemble the PDE for CO2 in the membrane. The electrochemical reaction at the Au
electrode appears in BCs (4.4.3), (4.7.3) and (4.8.3) in Table 4.
Figure 10 is a schematic of the sensor that is helpful when deriving the BCs. RAuI is the
radial distance to the inner edge of the gold electrode (in m) and RAuO is the radial
distance to the outer edge. R1 is the radial distance to the inner edge of the membrane and
R2 is radial distance to the outer edge of the membrane. Similarly, Z1 is the vertical
distance to the bottom of the membrane and Z2 is the vertical distance to the top surface
of the membrane. Some of the BCs are also indicated by numbers and arrows in Table 3
on the figure.
36
Figure 10: Schematic of pCO2 Sensor The upper (outer) surface of the gas-permeable membrane is in contact with the blood or
the calibrant fluid. At this outer edge of the membrane, the CO2 partial pressure in the
blood is assumed to be the same as the bulk CO2 partial pressure in the blood sample (see
Assumption 1.2 in Table 1). Concentration gradients within the blood sample are
neglected as is the depletion of CO2 in the blood over time due to the small volumes of
the membrane and electrolyte compared to the volume of the blood. At the outer
membrane surface in contact with blood, the partial pressure of CO2 is assumed to be in
equilibrium with the CO2 concentration just inside the membrane (at
, , , , which gives rise to BCs (4.1.1) and (4.1.4) in Table 4 where is
Henry’s constant in mmol/ kPa-1 m-3.
The CO2 concentration in the membrane at its inner surface is assumed to be in
equilibrium with the H CO in the neighboring electrolyte, which gives rise to BCs 4.1.2
and 4.1.5, for CO2 where is a partition coefficient. Initial condition (4.1.7)
indicates that the initial [CO2] in the membrane is in equilibrium with the initial
concentration of H2CO3 in the electrolyte [H2CO3]0.
37
BC 4.2.1 for H2CO3 at the upper surface of the electrolyte can be derived by considering
a ring with thickness Δr on the flat upper surface of the electrolyte, as shown in Figure
11. The number of moles of CO2 that diffuse into the membrane surface during a short
period of time Δt is equal to the number of moles of H2CO3 that diffuse from the surface
and into the electrolyte:
2 ∆
,∆ 2 ∆
,∆
(28)
Dividing by 2rΔrΔt gives BC 4.2.1 in Table 4. A similar argument is used to obtain BC
4.2.3.
Figure 11: Small ring with thickness Δr BCs for other species at the interface between the electrolyte and the gas-permeable
membrane are zero-flux conditions because none of the other species can diffuse into or
out of the membrane. BC (4.2.4) arises from the radial symmetry of the electrolyte,
which results in a minimum in [H2CO3] at r=0.
The bottom surface of the electrolyte consists of two different regions: the inert surface
and the Au electrode surface. For the purposes of this model, the Ag/AgCl electrode is
treated as an inert surface (see Assumptions 1.10 and 1.12 in Table 1).
38
For the inert portions of the bottom surface of the electrolyte, there is no diffusion of any
species in the z direction, resulting in BCs (4.3.2), (4.4.2), (4.5.2)., (4.6.2), and (4.7.2) in
Table 4.
At the Au electrode surface, H+ and BQ diffuse from the electrolyte onto the Au surface
to react with electrons supplied by the Au to produce H2Q according to chemical
equilibrium reaction (2), resulting in BCs (4.4.3) and (4.7.3). The H2Q that is produced
diffuses away from the Au electrode into the bulk electrolyte solution, resulting in BC
(4.8.3).
The potential of the Au electrode changes when BQ reacts with H+ and electrons as
described by Equation (7), resulting in a change in krAu with time, due to a change in the
equilibrium constant .Assuming that remains constant (See Assumption
1.17 in Table 1):
(29)
Since the forward and reverse reaction rates are very fast and the concentrations are in
equilibrium defined by which is changing in time and leading to changes in
the concentration ratio, we can assume that:
(30)
where the subscript Au indicates an average concentration over the surface of the Au
electrode, which is computed using the “Boundary Probe” function in COMSOLTM that is
used to compute the average voltage response for the forward and reverse BQ⇌ H2Q
reaction. Additional details on the implementation in COMSOLTM are in the next
39
section, 2.3.The updated potential, E, between the Au and Ag/AgCl electrode can then be
calculated from:
. .
31
which is obtained by rearranging Equation (10).
Initial concentrations for species in the electrolyte are determined from the known initial
concentration of NaHCO3 in the electrolyte recipe (see Table 2) and the known carbon
dioxide dissolved (H2CO3) in water at 25 oC which is 10-5 mol/L. Since half the water in
the electrolyte is assumed to evaporate and the concentration of H2CO3 during
manufacturing is constant, the known concentration of the buffer is doubled. The initial
concentrations of all species in the electrolyte (except for BQ and H2Q) can be
determined from:
(32)
2 2 (33)
Calculation of the initial conditions for BQ and H2Q requires Equation ( and the
following mass balance:
(34)
and the initial value of E is known in mV (APOC Company Confidential) given:
40
0.699 ln 0.22233 ln (35)
is the activity or effective concentration of chloride ion, which can be calculated
using (Newman and Thomas-Alyea, 2004):
(36)
where is the activity coefficient of chloride ion (Newman and Thomas-Alyea, 2004):
√√
(37)
= -1 is the charge number for Cl-, α is one of the Debye-Huckel parameters for aqueous
solutions in , B is the other Debye-Huckel parameter for aqueous solutions in ,
and a is the mean diameter of the hydrated chloride ion in nm.
I is the molal ionic strength of the solution in calculated from:
(38)
where I is the molar ionic strength of the solutions in and is the density
of the pure solvent in . The molar ionic strength is calculated from :
12
(39)
where is the charge number of species i and is the concentration of species i in .
41
Two Debye-Huckel parameters, α and B at 37 oC, are required to calculate I’. These
values were obtained by fitting a quadratic using values of the two Debye-Huckel
parameters at the temperatures listed below in Table 6 (Newman and Thomas-Alyea,
2004):
Table 6: Values of α from 273.15 K to 348.15 K.
T (K) 1 1 α ( B (
273.15 0.003661 1.1324 3.248 298.15 0.003354 1.1762 3.287 323.15 0.003095 1.2300 3.326 348.15 0.002872 1.2949 3.368
Curves of best fit were estimated using linear least squares regression:
1391171
1113.91
3.3461
569511
523.561
4.4017
as shown in Figure 12:
42
Figure 12: Curves of best fit for α and B.
giving α=1.2008 and B= 3.3057 at 37 ºC.
The value I = 0.98 shown in Table 2 was obtained from:
12
1 1 1 1
2 1 1
(40)
The only remaining unknowns in Equation (33) are [H2Q]0 and [BQ]0. As a result
Equation (33) can be combined with Equation (32) to obtain the initial conditions for the
quinones.
The ionic diffusivities listed above are for ions in pure water. The true diffusivities in the
electrolyte may be lower due to the fact that 10% of the electrolyte by weight is sucrose,
which increases the visocity of electrolyte solution. In order to maintain electroneutrality
y = 139117x2 ‐ 1113.9x + 3.3461
y = 56951x2 ‐ 523.56x + 4.4017
3.24
3.26
3.28
3.3
3.32
3.34
3.36
3.38
1.12
1.14
1.16
1.18
1.2
1.22
1.24
1.26
1.28
1.3
1.32
0.0028 0.003 0.0032 0.0034 0.0036 0.0038
α
1/T (1/K)
B
43
in the electrolyte, ions tend to diffuse in pairs. In conclusion, the diffusion of a cation
and its counter anion can be lumped together to obtain a reasonable diffusion coefficient
for the corresponding ion pair. For example, the most probable anion that will diffuse
with is because it is the most abundant anion in the electrolyte. There is about a
million time more ions than ions. Therefore, the ion pair diffusion coefficient
for H+ and should be approximately the lower value of their two diffusivities,
which is 1.48 10 m2/s. Similary, OH- should have approximately the same
diffusivity as in the solution because Na+ is the most abundant cation. Table 7 lists
the corresponding diffusivity parameters used in the simulations.
Table 7: Lumped Diffusion Coefficients Used in Model
Diffusivity Value
2.20 10 m2/s
2.70 10 m2/s
1.48 10 m2/s
1.79 10 m2/s
1.48 10 m2/s
D D 1.18 10 m2/s
2.3ModelImplementationinCOMSOLTM:COMSOLTM is a finite element method (FEM) tool for numerically solving PDEs. A
solution is computed by discretizing the domain into elements so that the PDEs can be
converted to algebraic equations. The density of the elements in the domain can have an
effect on both the accuracy of the solution and the computation time of the model. The
44
“user controlled mesh” was used in COMSOLTM and a maximum element size of 0.8 um
and a relative tolerance of 1 Χ 10-7 s for the time were used for the simulations (See
Appendix). The solver used in this thesis was the MUMPS (MUltifrontal Massively
Parallel sparse direct Solver). This uses a multifrontal method, which is a version of
Gaussian elimination for a large system of equations which arises from the finite element
method.
Since numerical solutions are approximate, a certain amount of error is associated with
the calculations. It is important to analyze the solution to determine the reliability of the
results. At the same time, it is important to consider the typical magnitudes of
measurement errors that would be encountered for the dependent variables, such as the
species concentrations and potential. In the discussion that follows, an overview is
presented of how the model was implemented in COMSOLTM to assist future users of the
model, and those wishing to reproduce the results. COMSOLTM version 4.3.0 was used
for the simulations in this thesis.
The COMSOLTM tree structure for the model implementation is shown in Figure 13.
Under the “Global Definitions” tab, there is a list of parameters (see Table 8) and a list of
variables (see Table 9) that appear in the model. Parameters such as the partition
coefficient , the equilibrium constant and the diffusion coefficient are
defined in the parameter list. Calculated variables such as ( in COMSOLTM
instead of for simplicity) and are defined in the variable list. At time zero,
when the calibrant fluid first contacts the membrane, is computed from Equation (33)
45
as a function of the known initial voltage response. Thereafter, is computed from
Equation (30).
Figure 13: Tree structure of the Transport of Diluted Species Modules
Table 8: Parameter List in COMSOLTM. Symbol Description Value Reference Diffusion coefficient of CO2 in
membrane at 37 oC 2.20 10 m2/s
(Yang and Kao, 2010)
Diffusion coefficient of at 37 oC
2.70 10 m2/s
(Zeebe, 2011)
Diffusion coefficient of at 37 oC
1.48 10 m2/s (Newman and Thomas-Alyea, 2004)
Diffusion coefficient of at 37 oC
1.79 10 m2/s (Newman and Thomas-Alyea, 2004)
Diffusion coefficient of at 37 oC
1.18 10 m2/s (Green and Perry, 2008)
Equilibrium constant of at 37 oC 6.30 10
(Snokeyink and Jenkins, 1980) (Haynes, 2012)
Equilibrium constant of at 37 oC 5.78 10
(Snokeyink and Jenkins, 1980) (Haynes, 2012)
Equilibrium constant of O at 37 oC 2.39 10
(Snokeyink and Jenkins, 1980) (Haynes, 2012)
Initial concentration of APOC Company Confidential
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Initial concentration of APOC Company Confidential
Initial concentration of APOC Company Confidential
Initial concentration of APOC Company Confidential
Initial concentration of APOC Company Confidential
Initial concentration of APOC Company Confidential
Initial concentration of APOC Company Confidential
Initial concentration of APOC Company Confidential
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Initial concentration of APOC Company Confidential
Initial concentration of APOC Company Confidential
H Henry’s constant at 37 oC 0.230
∙
(Burtis et al., 2006)
Partition coefficient at 37 oC 1 (Burtis et al., 2006)
Forward reaction rate constant at the Au electrode at 37 oC
600
∙
Estimated
Forward reaction rate constant for the dissociation of carbonic acid at 37 oC
1000 Assumed to be large
Forward reaction rate constant for the dissociation of bicarbonate at 37 oC
1000 Assumed to be large
Forward reaction rate constant for the dissociation of water at 37 oC
1000 Assumed to be large
Reverse reaction rate constant for the dissociation of bicarbonate at 37 oC
Due to equilibrium ratio
Reverse reaction rate constant for the dissociation of carbonic acid at 37 oC
Due to equilibrium ratio
47
Reverse reaction rate constant for the dissociation of water at 37 oC
Due to equilibrium ratio
Table 9: Variable List
As noted earlier, an average voltage response over the Au electrode surface is required
for reaction (7) and Equation (10) and this is accomplished using a COMSOLTM
Boundary Probe function that is specified in the “Definitions” tab under the “Model” tab,
as shown in Figure 14.
Figure 14: Voltage response at the Au electrode
48
49
Under the “Geometry” tab the layout of the domain in Figure 15 is specified using 2-D
axisymmetric settings. Note that the corresponding dimensions are specified as
parameters in Table 9. Nothing was defined under the “Material” tab because all
parameters relating to material characteristics, such as the partition coefficient and
diffusion coefficients, are defined in the parameter list.
PDEs as well as boundary and initial conditions were entered in COMSOLTM using the
Diffusion of Diluted Species Module in the library module. Two modules were created:
one for the gas-permeable membrane, and another for the electrolyte (see Figure 13). In
the Transport of Diluted Species (GPM) module that was defined (GPM stands for gas
permeable membrane) only one dependent variable was specified because only one
species diffuses within the membrane (i.e., CO2m). In the Transport of Diluted Species
(Electrolyte) module that was defined, seven dependent variables were specified because
there are seven PDE material balances for this part of the domain (i.e., balances on
H2CO3, HCO , H , OH ,CO ,BQ, and H2Q shown in Table 3). The Transport of
Diluted Species Module library automatically builds terms in the PDEs associated with
diffusion of each species, according to the specifications for each of the modules of this
type that are defined (e.g., GPM or Electrolyte). These are stored as Transport of Dilute
Species modules.
Under the Transport of Diluted Species modules there are sub-tabs as shown in Figure
15. Under the “Diffusion GPM” tab, the label for the diffusion coefficient,
, was entered. Similarly, diffusion coefficients for species in “Transport of
Diluted Species (Electrolyte)” were defined under the “Diffusion Electrolyte” tab.
50
Figure 15: Sub tabs in Transport of Diluted Species modules
51
Figure 16: Diffusion coefficients under “Diffusion GPM” and “Diffusion Electrolyte”
52
In Figure 15, “Axial Symmetry GPM” and “No Flux GPM” are default BCs where the
flux is 0 in the middle due to: radial symmetry (i.e., at r=0), at the walls (i.e., at z=0), and
on the sides of the domain (r=R2). Similarly, “Axial Symmetry Electrolyte” and “No
Flux Electrolyte” BCs are specified where appropriate according to the BCs shown in
Table 4. Initial conditions are included for each species by entering the corresponding
initial concentrations under the “Initial Values GPM” tab and “Initial Values Electrolyte"
tab in Figure 17.
53
.
Figure 17: Initial conditions in membrane and electrolyte
Reactions in the electrolyte were specified using the “Reactions Electrolyte” tab shown in
Figure 18
54
Figure 18: Reaction tab in electrolyte
Under the “Flux CO2” and “Flux H2CO3” tabs shown in Figure 19, special BCs such as
BC (4.2.1) in Table 4 were specified using the Flux function where the syntax
“ndflux_xx” was used to specify the rate of diffusion of the species. The symbol “+”
indicates that H2CO3 is entering the electrolyte. The value of GPM.ndflux_C_CO2 is
computed by COMSOLTM using the PDEs.
55
Figure 19: Continuous flux at the interphase between gas-permeable membrane and electrolyte
Under the “CO2 Concentration” and “H2CO3 Concentration” tabs in Figure 20, BCs
involving the known concentration at a boundary (such as BCs (4.1.1) and (4.1.4)) were
implemented using the “Concentration” function.
56
Figure 20: BCs involving the Henry’s constant. BCs involving the partition coefficient between the membrane and the electrolyte (i.e.,
BCs (4.1.2) and (4.1.5) in Table 4) were implemented as shown in Figure 21. Note that
the Concentration tab in Figure 20 was opened from the Transport of Diluted Species
(GPM) tab in Figure 15 and that the Concentration tab in Figure 21 was opened from the
Transport of Diluted Species (Electrolyte) tab in Figure 15.
57
Figure 21: BCs involving the partition coefficient between the membrane and the electrolyte
Equ
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59
from constitutive relationships such as the Nernst equation, initial and BCs, and the
implementation in COMSOLTM, using library modules such as the Transport of Dilute
Species, and probes and flux calculations.
The model is fitted in the following chapter, and laboratory test data are used to adjust
several key parameters in order to obtain predictions that match observed behavior. The
fitted model is then used to conduct a series of sensitivity investigations, in which the
impact of key design parameters on predicted sensor response is determined.
60
3.1OverviewThe mathematical model for the pCO2 sensor was solved using COMSOLTM, and fitted
with a number of datasets obtained from an Abbott Point of Care product testing
database. Four sets of data were obtained for POC pCO2 sensors using the calibrant fluid,
and for four different control fluids with observed pCO2 values, as shown in Table 10
(Abbott Point of Care-d, 2012). The unit mmHg was used due to the fact the data was
collected under this unit. It was converted to kPa in COMSOLTM.
Table 10: pCO2 Values of Calibrant and Control Fluids.
Calibrant and Control Fluids Observed pCO2 Values (mmHg)
Calibrant fluid APOC Company Confidential
CV1 89.4
L2 29.3
GB 22.2
CV5 17.8
Each run consists of a calibrant/control fluid pair in which the first portion of the run was
with a standard calibrant (contained within the test cartridge), followed by a control fluid
(entered into the cartridge by an operator). Note that CV1, L2 and CV5 are control fluids
(aqueous solutions) and GB (glucose blood) is blood collected from donors infused with
a known level of CO2. This blood is collected by the phlebotomist at Abbott Point of
Care and its pCO2 value is determined offline right before usage. Voltage versus time
data was collected for each fluid for four replicate experiments. Representative plots of
voltage versus (vs) time for each fluid are plotted in Figure 23 (the time has been
normalized for the time axis). Note that the green error bars (corresponding to one
61
standard deviation of measurement noise) are shown on the (overlapping) symbols to
illustrate the reproducibility of the calibrant and control-fluid data (which were all
collected in October, 2012). These plots have an approximately linear slope during the
time periods when the calibration fluid or a control fluid is in contact with the membrane
(Cai and Reimers, 1993).
a)
62
b)
c)
63
d)
Figure 23: Voltage responses as a function of time for calibrant (from 1.15 to 1.48) and control fluids (from 1.57 to 2.32) for a) CV1, b) L2, c) GB, and d) CV5. Green error bars for measurements are show at two time points to illustrate reproducibility.
3.2SimulatedVoltageResponsePlotsforCalibrantA voltage versus time plot was generated in COMSOLTM using an input pCO2 value of
the calibrant, as specified in Table 2. The results are shown in Figure 24. Note that these
simulation results were obtained using the initial conditions and the parameter values in
Table 8. Parameter kfAu, was adjusted to obtain a good match between the simulation
results (in black) and the data values in red. No data were available before t = t2 (see
Figure 4 and associated discussion). Note that the manually adjusted value kfAu = 666.28
∙ is the initial parameter value reported in Table 10 because no literature values for
64
this parameter were available and it was determined using these preliminary simulation
results. Simulations with alternative values of kfAu are shown in Figures 25 and 26.
Figure 24: Voltage response for a simulation for calibrant in COMSOLTM with kfAu =
666.28 ∙
Figure 25 is a simulation for a kfAu value of 160 ∙
and Figure 26 is a simulation for a
kfAu value of 700 ∙
. kfAu changes the slope of the voltage versus time plot, with
higher kfAu values yielding steeper slopes. Note that when kfAu is adjusted, the reverse rate
constant krAu is automatically recalculated via the equilibrium constant, so that Equation
(28) is satisfied. Figures 25 and 26 confirm that the value of kfAu = 666.28 ∙
provides
the best fit.
Figure 27 shows the initial voltage response of all control fluids, revealing a mismatch
between the simulated results and the data, indicating that one or more parameters need
to be adjusted to obtain a good fit to the data. Unfortunately, the version of COMSOLTM
65
that was used did not have any parameter estimation capabilities or a Matlab Livelink so
that Matlab could be used to estimate the parameters. Rather than conducting a formal
nonlinear least-squares parameter estimation, a few key parameters were identified and
adjusted manually to improve the fit to the data. Abbott Point of Care uses a mean-based
algorithm that calculates the average voltage response between a predetermined number
of seconds (APOC Company Confidential) for the control fluid run. This average voltage
response value corresponds to the pCO2 in each of the control fluids. This value can be
compared to the known pCO2 value (calibrant) in order to generate a quantitative result.
Figure 25: Voltage response for a simulation with kfAu = 160 ∙
66
Figure 26:Voltage response for a simulation with kfAu = 700
∙
Figure 27: Voltage response for all control fluids. The red marks are data points from experiments. The solid red marks are the initial predetermined number of seconds that are used to calculate the mean voltage response.
67
3.3IdentifyingtheTwoMostInfluentialParametersThe 11 parameters that could be considered for estimation are: , , ,
, , , H, , , , , . An estimability ranking procedure
was used to determine that parameters and are the most influential (relative
to their uncertainties) (Thompson et al., 2009; McLean and McAuley, 2012). The
estimability analysis is conducting using first-order sensitivity information (first
derivative information) that describes the impact of perturbations of parameters on the
predicted responses, with all other parameters held constant at nominal values. This
sensitivity information can be generated analytically if algebraic models are available. In
this instance, because the model consisted of PDE and algebraic equations requiring
numerical solution using COMSOLTM, a finite-difference perturbation approach was used
to determine the first-order sensitivity information. Each of the 11 parameters was
perturbed one at a time (by 10% of their nominal values), and the solution trajectories
were computed using COMSOLTM. The response being considered was the average
voltage at the surface of the Au electrode for a predetermined number of seconds (APOC
Company Confidential). This response was considered because of the algorithm used in
Abbott Point of Care. A mean-based algorithm is employed in the pCO2 sensor design.
In order to apply the estimability algorithm of McAuley and co-workers, the scaled
sensitivity matrix corresponding to the runs being considered has to be formed. In this
instance, the sensitivities were considered over the four control fluid runs identified
earlier (CV1, L2, GB and CV5), which makes 4 rows in the scaled sensitivity matrix.
Since there are 11 parameters, the sensitivity matrix is a 4 X 11 matrix whose elements
are:
68
∆
∆
where∆ is the resulting change in predicted average voltage (for fluid j) when
parameter is adjusted by ∆ . is the uncertainty in the initial value of the ith
parameter, which is calculated as half the distance between the lower and upper bound
for the parameter (see Table 2). is the uncertainty in the measured average voltage,
which was assumed to be the same for all four fluids. This value = 2.48 mV was
determined using the pooled standard deviation of the average voltage responses from
five sets of dynamic experiments for each control fluid. The elements of the resulting
scaled sensitivity matrix (to two decimal places of accuracy) are shown in Table 11.
These elements are dimensionless due to the scaling that was used.
The estimability algorithm of McAuley and co-workers determines the most influential
parameters, after taking into account the co-dependencies in the impact of parameters on
the predicted response. This is accomplished using an orthogonalization algorithm
(Thompson et al., 2009; McLean and McAuley, 2012). The algorithm determined that
parameters kfAu and κCO2m are the most influential parameters. This result is not
surprising because of the relatively large magnitudes of the scaled sensitivity entries in
columns 1 and 5 in Table 11 compared to the entries in columns corresponding to the
other parameters.
69
Table 11: Scaled Sensitivity Matrix for Parameter Ranking Fluids H
CV1 70.86 0.01 0.00 0.00 20.69 0.00 -0.02 -0.03 0.00 0.00 -1.53
L2 9.70 0.01 0.00 0.00 2.82 0.09 0.00 0.00 0.00 0.00 -0.11
GB 10.05 0.01 0.00 0.00 3.44 0.11 0.00 0.00 0.00 0.00 -0.08
CV5 5.45 0.01 0.00 0.00 1.59 0.05 0.00 0.00 0.00 0.00 -0.03
70
The two most estimable parameters were hand tuned in order to improve the predictions
of the average voltage response. Adjustments were introduced, and COMSOLTM was
used to generate new trajectories from which average voltage response values were
determined. The resulting “best” values of the two most influential parameters were
= 66.6 ∙
and κ =1.2. Predictions using the tuned parameter values are
shown in Figures 28 and 29. A good match with the data was obtained over the
predetermined number of seconds (solid symbols). However, the simulated data did not
fit the data as well after the predetermined number of seconds (open symbols) for the L2
and CV5 runs. The experimental data for L2 has a steeper slope compared to the rest of
the control fluids. Note that a better fit to the data might be obtained using least-squares
parameter estimation instead of empirical hand-tuning.
In order to have a better understanding of the underlying phenomena in the sensor, a
number of “sectioning” plots were generated to study concentration profiles along the
vertical and radial axes. The resulting plots are summarized for a number of key species.
Concentration profiles for carbonic acid within the calibration fluid were simulated at
several different times in Figure 30 using the tuned parameter values and conditions that
correspond to the run shown in Figure 31. These vertical profiles were determined at the
midpoint of the Au electrode annulus (r = 1.265 10-4 m), as shown in Figure 32. Note
that because there is no angular variation, this profile would be the same anywhere along
the midpoint radius on the Au electrode annulus. At time zero, the concentration profile
is flat and corresponds to the initial concentration indicated in Assumption 1.2 in Table 1.
When the calibration fluid comes in contact with the membrane, CO2 begins to diffuse
71
through the membrane and to dissolve as H2CO3. After 0.1 ms, the additional H2CO3 has
penetrated about 1.4 µm into the electrolyte and by 100 ms, the vertical concentration
profile is uniform. These simulation results suggest that the dynamics of the diffusion
within these small POC sensors are very fast compared to the dynamics in larger
Severinghaus sensors described in the introduction. Furthermore, these results suggest
that stable voltage readings could be obtained in an even shorter period of time than is
used in the current sensor. The main impediment to achieving a fast response may be
heating of the sensor (i.e., the sensor can be heated faster to 37 oC so the voltage response
reading can be generated sooner) rather than diffusion of CO2.
Figure 28: Mean voltage response vs pCO2 plots for the base and tuned cases
72
Figure 29: Voltage response for all control fluids for using initial parameter values in Table 8 are represented by dashed lines and voltage response using tuned parameter values are represented by solid lines.
Figure 30: Concentration profiles for carbonic acid at the middle of the Au surface.
73
Figure 31: Voltage response for calibrant used for concentration profiles at centre of the Au electrode
Figure 32: Centre of Au electrode where carbonic acid concentration profiles are taken
3.4SensitivityAnalysis Part of the value of a mathematical model of the POC sensor response is the ability to
study the impact of design decisions on the performance of the sensor. In part as proof of
concept, and as a preliminary step in using the model to elucidate the impact of design
decisions on sensor behavior, a number of design parameters in the sensor were varied to
observe their influence on the voltage response. Sensitivity analyses were performed to
determine the most sensitive design parameters in the sensor, over the ranges of practical
interest. The factors that were studied are: water concentration in the electrolyte, height
74
of the electrolyte within the sensor (because this will vary depending on the water that
has evaporated), initial dissolved CO2, initial buffer concentration and initial BQ in the
electrolyte composition. Note that with a fixed cross-sectional area, varying the height of
electrolyte is equivalent to varying the volume of electrolyte in the sensor. The
corresponding sensitivity plots are shown in Figures 33 to 46. Note that only the results
for CV1 and CV5 are shown. These fluids have the highest and lowest pCO2 values,
respectively, of the four control fluids, and were used to bracket the range of pCO2 values
that would typically be encountered.
Figure 33 depicts the voltage response for the calibrant when different amounts of water
are present in the electrolyte, while the height and hence volume of the electrolyte is held
at its nominal design value. The voltage increases by 40% at 0.88 when more water is
present. Similarly, the voltage decreases by 35% at the same time when less water is
present. Figure 34 presents the voltage response for the control fluids under the same
scenario. The predicted voltage is significantly higher for both CV1 and CV5 when there
is more water in the electrolyte. Similarly, the voltage is lower when there is less water
in the electrolyte. These results make sense because the initial concentration of H2Q at
the electrode surface is lower while the initial concentration of H+ is higher when there is
more water. Both BQ and H2Q concentrations are reduced when there is more water.
However, the H+ concentration is increased when there is less water due to Equations
(32) and (33) and equilibriums (17) to (19). The concentrations of H2Q and H+ dominate
the voltage response in Equation 7. When there is a lower concentration of H2Q and a
higher concentration of H+, the magnitude of ln decreases, which increases
the value of due to the subtraction of a lower value of ln in
75
Equation (7). A higher value of leads to a higher value of E at the Au electrode
due to Equation (8) since Eref is constant. Also, when there is more H+ present at the Au
electrode surface, the cations attract more electrons in the Au electrode surface,
increasing the accumulated charge separation and yielding a more positive voltage
response.
Figure 33: Influence of water concentration in electrolyte voltage response for the calibrant. The thickest lines are for the case with the most water (3/4 of the amount in as-manufactured electrolyte). The medium thickness lines are for the base case (1/2 of the amount in as-manufactured electrolyte) and the thinnest lines are for the case with the least water (1/4 of the amount in as-manufactured electrolyte).
76
Figure 34: Influence of water concentration in electrolyte on voltage responses are shown for CV1 , and CV5 - - - . The thickest lines are for the case with the most water (3/4 of the amount in as-manufactured electrolyte). The medium thickness lines are for the base case (1/2 of the amount in as-manufactured electrolyte) and the thinnest lines are for the case with the least water (1/4 of the amount in as-manufactured electrolyte).
Figures 35 and 36 depict the voltage responses when the height of the electrolyte
changes. There is no noticeable change in the voltage response for both the calibrant and
control fluids. A thicker or thinner electrolyte does not appear to affect the voltage
response. The diffusion coefficients of all species are fast compared to the thickness of
the membrane. Therefore, the membrane has to be extremely thick (i.e., in the millimeter
range) or extremely thin (i.e., in the nanometer range) in order to make a difference.
Having an extremely thick membrane drives up the cost. Therefore it is undesirable.
Having an extremely thin membrane is likely to prove challenging for consistent and
robust manufacturing efforts.
77
Figure 35: Influence of the electrolyte height on voltage response for the calibrant. The thickest lines are for the case with the most water (3/4 of the amount in as-manufactured electrolyte). The medium thickness lines are for the base case (1/2 of the amount in as-manufactured electrolyte) and the thinnest lines are for the case with the least water (1/4 of the amount in as-manufactured electrolyte).
Figure 36: Influence of water concentration in electrolyte on voltage responses are shown for CV1 , and CV5 - - - . The thickest lines are for the case with the most water (3/4 of the amount in as-manufactured electrolyte). The medium thickness lines are for the base case (1/2 of the amount in as-manufactured electrolyte) and the thinnest lines are for the case with the least water (1/4 of the amount in as-manufactured electrolyte).
78
Figure 37 illustrates the voltage response for the calibrant when the amount of water and
height of the electrolyte are changed together. This produces the same results as those
shown in Figure 33, since the height of the electrolyte has no effect on the voltage
response as shown in Figures 35 and 36. Consequently, the voltage response for the
control fluids shown in Figure 38 is the same as for the case of changes in the amount of
water, shown in Figure 34. Therefore, only the amount of water present in the electrolyte
affects the voltage response, regardless of the electrolyte geometry.
Figure 37: Influence of water concentration in electrolyte and electrolyte thickness on voltage response for the calibrant. The thickest lines are for the case with the most water and thickest electrolyte (3/4 of the water amount and 1.5X the thickness in as-manufactured electrolyte). The medium thickness lines are for the base case (1/2 of the amount and original thickness in as-manufactured electrolyte) and the thinnest lines are for the case with the least water (1/4 of the water amount and 1/2X the thickness in as-manufactured electrolyte).
79
Figure 38: Influence of water concentration in electrolyte on voltage responses are shown for CV1 , and CV5 - - - . The thickest lines are for the case with the most water and thickest electrolyte (3/4 of the water amount and 1.5X the thickness in as-manufactured electrolyte). The medium thickness lines are for the base case (1/2 of the water amount and original thickness in as-manufactured electrolyte) and the thinnest lines are for the case with the least water (1/4 of the water amount and half of the thickness in as-manufactured electrolyte).
The voltage response for the calibrant when different levels of carbonic acid are present
in the electrolyte is shown in Figure 39. The sensor voltage response is increased by 5%
relative to the nominal design when there is more carbonic acid present. Similarly, the
sensor voltage response is lower relative to the nominal design when there is less
carbonic acid. Figure 40 contains the voltage response for the control fluids when
different levels of carbonic acid are present in the electrolyte. The sensor voltage
response is increased by ~5% for both CV1 and CV5 when there is more carbonic acid
present. Similarly, the sensor voltage decreases by about ~5% for both control fluids
when there is less carbonic acid. This makes sense physically, since at higher levels of
carbonic acid, there is more H+ present at the Au electrode surface which attracts more
80
electrons in the Au electrode surface, increasing the accumulated charge separation and
yielding a more positive voltage response.
Figure 39: Influence of initial carbonic acid concentration in electrolyte on voltage response for the calibrant. The thickest lines are for the case with the most carbonic (twice of the amount in as-manufactured electrolyte). The medium thickness lines are for the base case (the amount in as-manufactured electrolyte) and the thinnest lines are for the case with the least carbonic acid (half of the amount in as-manufactured electrolyte).
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Figure 40: Influence of initial carbonic acid concentration in electrolyte on voltage responses are shown for CV1 , and CV5 - - - . The thickest lines are for the case with the most carbonic (twice of the amount in as-manufactured electrolyte). The medium thickness lines are for the base case (the amount in as-manufactured electrolyte) and the thinnest lines are for the case with the least carbonic acid (half of the amount in as-manufactured electrolyte).
The voltage response for the calibrant when different levels of the buffer are present in
the electrolyte solution is shown in Figure 41. The voltage response is increased by ~25%
relative to the nominal design when the amount of buffer is doubled, and it is reduced by
about ~25% when the amount buffer is halved. The corresponding voltage profiles for the
control fluids are shown in Figure 42, at different levels of the buffer present in the
electrolyte solution. The voltage response increases by ~10% when the amount of buffer
is doubled, and the voltage response decreases by ~10% when the amount is halved for
CV1. Similarly, the voltage response increases by ~10% when the amount of buffer is
doubled, and the voltage response decreases by ~10% when the amount is halved for
CV5. This makes sense physically, since when more buffer is present, there is less H2Q
per volume initially present at the Au electrode surface which yields a higher voltage
response. This agrees with Figures 33 and 34.
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Figure 41: Influence of initial buffer concentration in electrolyte on voltage response for the calibrant. The thickest lines are for the case with the most buffer (four times of the amount in as-manufactured electrolyte). The medium thickness lines are for the base case (twice of the amount in as-manufactured electrolyte) and the thinnest lines are for the case with the least buffer (the amount in as-manufactured electrolyte).
83
Figure 42: Influence of initial buffer concentration in electrolyte on voltage responses are shown for CV1 , and CV5 - - - . The thickest lines are for the case with the most buffer (four times of the amount in as-manufactured electrolyte). The medium thickness lines are for the base case (twice of the amount in as-manufactured electrolyte) and the thinnest lines are for the case with the least buffer (the amount in as-manufactured electrolyte).
Changes in the amount of BQ present while maintaining the other species concentrations
at their nominal design values produce no change in the voltage response for the
calibrant, or for the control fluids. The initial concentration ratio of H2Q to BQ is still the
same. Therefore there is no change in the voltage. The voltage responses for the calibrant
are shown in Figure 43, while those for the control fluids are shown in Figure 44.
Finally, changing the area of the Au electrode surface does not affect the voltage
84
response for either the calibrant or control fluids, as shown in Figures 45 and 46.
Figure 43: Influence of initial BQ concentration in electrolyte on voltage response for the calibrant. The thickest lines are for the case with the most BQ (four times of the amount in as-manufactured electrolyte). The medium thickness lines are for the base case (two times of the amount in as-manufactured electrolyte) and the thinnest lines are for the case with the least BQ (the amount in as-manufactured electrolyte).
Figure 44: Influence of initial BQ concentration in electrolyte on voltage responses are shown for CV1 , and CV5 - - - . The thicker lines are for the case with the most water (amount in as-manufactured electrolyte). The regular thickness lines are for the base case (twice of the amount in as-manufactured electrolyte).
85
Figure 45: Influence of Au electrode surface area voltage response for the calibrant. The thickest lines are for the case with the biggest Au electrode area (twice of the surface area in as-manufactured Au electrode). The medium thickness lines are for the base case (the surface in as-manufactured Au electrode) and the thinnest lines are for the case with the least water (half of the surface area in as-manufactured Au electrode).
Figure 46: Influence of Au electrode surface on voltage responses are shown for CV1 , and CV5 - - -. The thicker lines are for the case with the most water (twice of the Au electrode surface area in as-manufactured). The regular thickness lines are for the base case (the Au electrode surface area in as-manufactured).
Summarizing, the amount of water in the electrolyte is the most influential design factor
for the voltage response. The sensor voltage response increases relative to that of the
nominal design when there is more water, and decreases when there is less water. In
86
contrast, the voltage response is not influenced by changes in the height of electrolyte,
BQ concentration, or surface area of the Au electrode.
The insights from these sensitivity analyses can be used in several ways. First, the
sensitivity, or lack thereof, to changes in design parameters provides an indication of the
robustness of the performance to variations in the design parameters. Changes in the
amount of water in the electrolyte have a more pronounced effect, suggesting that water
content in the electrolyte is a key factor to monitor, relative to other characteristics such
as the height or volume of electrolyte.
The second use for insights from these analyses is the possibility of modifying the design
of the sensor to produce a more dramatic or more rapid sensor response, enabling more
rapid measurement of pCO2, or modifications to the design to improve robustness or
reduce manufacturing cost.
3.5SummaryThe mathematical model of the POC pCO2 sensor has been successfully fitted using
sensor performance data from the Abbott Point of Care database. The most influential
model parameters influencing the predicted responses were identified using estimability
analysis, and values for the two most estimable parameters were tuned to improve the
quality of the model predictions. Finally, the use of the model for design investigation
was illustrated using a series of sensitivity analyses on the main design parameters for the
sensor. The model clearly indicated design factors having a pronounced influence on the
87
voltage response (e.g., water content), and those having a negligible influence (e.g., area
of the Au electrode).
88
Chapter4ConclusionsandRecommendations
4.1Conclusions In this thesis, a dynamic mathematical model was derived to predict the voltage response
in a pCO2 sensor when it is subjected to different CO2 concentrations in the calibration
fluid and control fluids. The model considers diffusion of species due to concentration
gradients in the vertical and radial directions, and reaction phenomena in the electrolyte
and at the Au electrode. Diffusion due to potential gradients and any potential change in
the reference Ag/AgCl were not considered.
The model predicts the two most influential and uncertain parameters in the model were
determined to be and , which are the forward rate constant for benzoquinone
consumption at the gold surface, and the partition coefficient for CO2 between the
membrane and the electrolyte. These parameters were adjusted by hand to obtain a good
fit (within 2 mV) between the dynamic voltage response data (during a predetermined
number of seconds) and the model predictions. An even better fit would be expected if a
formal least-squares parameter estimation study were performed or if additional
parameters were estimated.
The model was fitted using test data from Abbott Point of Care, and was found to provide
reliable predictions of the sensor voltage response over a time interval of interest.
Several design parameters were varied to study the influence of the electrolyte
concentration and the sensor geometry on the voltage response. The model predicts the
89
most influential design parameter studied was the amount of water present in the
electrolyte during the sensor operation. For example, increasing the water concentration
by 50% resulted in an increase of 35% in the predicted voltage for the calibrant when the
sensor is in contact with the calibrant fluid and an increase of 28% when the sensor is in
contact with CV1 control fluid. These results suggest that the amount of water that
evaporates or is absorbed by the sensor during manufacturing and storage may have an
important influence on the sensor response. The model predicts that the changing the
depth of the electrolyte fluid in the sensor was not as important as changing the water
concentration.
The model predicts the initial buffer concentration in the electrolyte was the second most
influential parameter. For example, increasing the buffer concentration from 0.005 mol
L-1 to 0.01 mol L-1 increased the predicted voltage by ~20% when the sensor is in contact
with the calibrant fluid and by ~10% when the sensor is in contact with CV1. The initial
H2CO3 concentration, which might depend on the CO2 concentration in the air during
manufacturing was the third most sensitive parameter. For example, increasing the
carbonic acid concentration from 0.00005 mol L-1 to 0.0001 mol L-1 increased the
predicted voltage by ~5% when the sensor is in contact with the calibrant fluid and when
the sensor is in contact with CV1. The model predicts that the initial benzoquinone
concentration in the electrolyte had very little influence on the sensor response, and the
surface area of the Au electrode was also not important to the predicted sensor response.
90
These results demonstrate the potential value of the mathematical model for providing
insight into influential design parameters.
4.2ContributionsofthisThesisThis thesis has advanced the mathematical modeling of POC pCO2 sensors beyond the
current state of the art in the literature. The specific contributions of the thesis are:
1. Development of the model equations and approach for dealing with the BQ⇌HQ
cascade and interaction at the electrode. This includes linking the Nernst equation
to the reaction equilibrium constant for the BQ⇌HQ reactions, and developing a
technique for initializing the concentrations in the sensor so that the PDE model
can be solved.
2. Development of the material balance PDE model for the membrane and
electrolyte, and subsequent parameter estimation using Abbott Point of Care test
data.
3. Gathering and grouping model parameters, including identifying which
characteristics (e.g., diffusivities) can be assumed to be similar.
4. An estimability analysis that identifies the most influential parameters in the PDE
model, enabling tuning to produce more reliable predictions.
5. A preliminary investigation into the impact of different design parameters on the
performance of the pCO2 sensor.
6. Identifying appropriate model routines and techniques within COMSOLTM to
represent, accommodate and solve the type of PDE model developed for the
sensor (e.g., computing an average potential over the Au electrode).
91
4.3RecommendationsforFutureWork1. The model would benefit from additional data that could serve to further validate the
predictions. Currently there are no plans to design and execute the experimental runs that
would generate this data.
2. Consideration should be given to constructing a more complicated PDE model to
account for changes in water concentration and temperature during the sensor operation.
Additional data would be required to fit the parameters in this model and to test the
model validity.
3. The diffusion of species within the electrolyte may be affected by the potential
gradient and by activities rather than the concentrations of the species alone. A more
complicated model that accounts for these effects should be developed, but would require
additional knowledge of transport phenomena and thermodynamics in solutions with
ions.
4. The potential of the reference electrode, Ag/AgCl, may change with time due to
changes in ionic strength and changes in species activities. This effect should be
incorporated in the model if additional data and knowledge are available.
5. The modeling approach should be applied to other POC sensors based on ion-selective
electrodes. The mathematical model structure consists of diffusing species, possible
dissociation in the electrolyte (depending on the analyte), and interaction with a
electrochemical reaction cascade, leading to the potential measurement at the electrode.
92
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Appendix:The “user-controlled mesh” was used in COMSOLTM 4.3.0. to generate the mesh and the
elements in the mesh are triangular (See Figure A.1). The number of iterations is 8 and
the integration order for the average voltage at the Au electrode is 4th order (See Figure
A.2).
Figure A. 1: “User-controlled mesh” in COMSOLTM 4.3.0
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Figure A. 2: Settings for number of iterations and integration order The maximum element size was varied (See Figure A.3) while keeping the other element
size parameters constant (i.e., minimum element size, maximum element growth rate,
resolution of curvature, and resolution of narrow regions). Maximum element sizes of
0.4, 0.6 and 0.8 um were selected. They each generated 16911, 8460 and 4724 elements
over the model domain respectively. The computation times for these three meshes were
about 40 min, 20 min and 10 min respectively.
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Figure A. 3: “Maximum element size” under “Element Size” tab
Figure A.4 is a fine mesh for the entire sensor with a maximum element size of 0.6 um.
Figures A.5 to A.7 are the meshes for one end of the sensor with maximum element sizes
of 0.4 um, 0.6 um and 0.8 um respectively.
.
Figure A. 4: Mesh for the central portion of the sensor (between r =0 m and r = 2.000
m) with a maximum element size of 0.6 um.
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Figure A. 5: Mesh for the outer portion of the sensor (between r = 2.670 10 m and r = 2.816 10 m) with a maximum element size of 0.4 um.
Figure A. 6: Mesh for the outer portion of the sensor (between r = 2.670 10 m and r = 2.816 10 m) with a maximum element size of 0.6 um.
Figure A. 7: Mesh for the outer portion of the sensor (between r = 2.670 10 m and r = 2.816 10 m) with a maximum element size of 0.8 um.
Parameter values in Table 8 plus updated values of = 66.6 ∙
and κ =1.2
were used to generate the following plots in Figures A. 8 and A.9 for the calibrant.
100
Since all the voltage vs time plots overlap and the concentration profiles overlap with
different maximum element sizes, grid independence was achieved. Therefore, a
maximum element size of 0.8 um was used.
Figure A. 8: Voltage responses versus time for maximum element size of 0.4 um, 0.6 um and 0.8 um. The three plots overlap.
Figure A. 9: Concentration profiles of carbonic acid for maximum element size of 0.4 um, 0.6 um and 0.8 um. The plots overlap.
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Time tolerance was also investigated. Under the “Time Dependent” tab (See Figure A.
10), the relative tolerance was varied. Three tolerances were used: 5X10-8 s, 2X10-7 s and
1X10-7 s. The same plots (See Figures A.11 and A.12) were generated and the voltage
plots overlap and the conentration profiles overlap as well. A relative tolerance of 1X10-7
s for the time setting was used for the simulations.
Figure A. 10: Setting up the relative tolerance for time in COMSOLTM.
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Figure A. 11: Voltage responses versus time for time relative tolerances of 5X10-8 s, 2Χ10-7 s and 1X10-7 s. The three plots overlap.
Figure A. 12: Concentration profiles of carbonic acid for time relative tolerances of 5 Χ 10-8 s, 2X10-7 s and 1Χ10-7 s. The plots overlap.
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‐135
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‐125
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69 71 73 75 77 79
V (mV)
t (s)